Dynamical systems under constant organization I. Topological analysis of a family of non-linear differential equations —A model for catalytic hypercycles

AbstractThe paper presents a qualitative analysis of the following systems ofn differential equations: $$\dot x_i = x_i x_j - x_i \sum\nolimits_r^n { = 1} x_r x_s {\mathbf{ }}(j = i - 1 + n\delta _{i1} {\mathbf{ }}and{\mathbf{ }}s = r - 1 + n\delta _{r1} )$$ , which show cyclic symmetry. These dynamical systems are of particular interest in the theory of selforganization and biological evolution as well as for application to other fields.